Lowest Energy State of Hydrogen Atom: Find A, Don't Miss the Obvious!

In summary, the hydrogen atom is in the state Ar^2e^{-r/a}cos(\theta). The quantum number l is 1, and the quantum number m is 0. The e^{(-r/a)} made the atom think that n=1, but the r^2 threw that out. The atom is in a lowest energy state, but the instructor wants the student to calculate the inner products first to ensure accuracy.
  • #1
Matthollyw00d
92
0
A hydrogen atom is in the state [tex]\psi=Ar^2e^{-r/a}cos(\theta)[/tex].
I need to find lowest energy state and etc. Obviously normalize to find A, but I'm not seeing the obvious linear combination of wave functions; and I really don't think my instructor wants me to do several inner products (plus actually calculate the several different wave functions) just to find the coefficients. Am I over looking something obvious?
 
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  • #2
You should be able to at least identify the quantum numbers l and ml. Start there.
 
  • #3
You should be able to figure out which spherical harmonic is involved pretty quickly. This gives you two quantum numbers. The degree of the radial polynomial tells you the largest value of n involved. The rest involves looking at a table of solutions and finding the right linear combination.
 
  • #4
That was my first instinct, which gave me l=1, m=0. But the [tex]e^{(-r/a)}[/tex] made me think that n=1, which then threw out my l (since l<n). Also the [tex]r^2[/tex], made me think n=2, but in either case I didn't see any good linear combination jumping out at me.(and it definitely wasn't just 210) I'll look at it again after I've had some coffee... it's been a long few days.
 
  • #5
I think you're going to have to calculate the inner products you were trying to avoid. If you use specific eigenfunctions, it's actually not too bad since all the integrands are just a polynomial multiplied by an exponential. You may be able to work out the general case, relying on some properties of the generalized Laguerre polynomials, like their orthogonality.
 
  • #6
Yah, I ended up using inner products and it all worked out. It's not that the problem was hard, the method just didn't seem of appropriate length. My professor is inconsistent with homework trends.
 

1. What is the lowest energy state of a hydrogen atom?

The lowest energy state of a hydrogen atom is known as the ground state, where the electron is in its most stable and lowest energy level around the nucleus.

2. How is the lowest energy state of a hydrogen atom calculated?

The lowest energy state of a hydrogen atom can be calculated using the Schrödinger equation, which takes into account the energy of the electron and its position in relation to the nucleus.

3. Why is finding the lowest energy state of a hydrogen atom important?

Finding the lowest energy state of a hydrogen atom is important because it helps us understand the fundamental principles of quantum mechanics and the behavior of atoms and molecules.

4. Can there be multiple lowest energy states for a hydrogen atom?

No, there can only be one lowest energy state for a hydrogen atom. This is because the electron can only occupy a specific energy level around the nucleus, and the ground state is the lowest energy level.

5. What is the "obvious" that shouldn't be missed when finding the lowest energy state of a hydrogen atom?

The "obvious" that shouldn't be missed when finding the lowest energy state of a hydrogen atom is the principle that electrons occupy the lowest energy levels first before moving to higher energy levels. This helps in determining the ground state of the hydrogen atom.

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